Plane posets, special posets, and permutations

We study the self-dual Hopf algebra HSPHSP of special posets introduced by Malvenuto and Reutenauer and the Hopf algebra morphism from HSPHSP to the Hopf algebra of free quasi-symmetric functions FQSym given by linear extensions. In particular, we construct two Hopf subalgebras both isomorphic to FQSym; the first one is based on plane posets, the second one on heap-ordered forests. An explicit isomorphism between these two Hopf subalgebras is also defined, with the help of two combinatorial transformations on special posets. The restriction of the Hopf pairing of HSPHSP to these Hopf subalgebras and others is also studied, as well as certain isometries between them. These problems are solved using duplicial and dendriform structures.